received: 01 June 2016 accepted: 05 September 2016 Published: 22 September 2016

Current driven spin–orbit torque oscillator: ferromagnetic and antiferromagnetic coupling Øyvind Johansen & Jacob Linder We consider theoretically the impact of Rashba spin–orbit coupling on spin torque oscillators (STOs) in synthetic ferromagnets and antiferromagnets that have either a bulk multilayer or a thin film structure. The synthetic magnets consist of a fixed polarizing layer and two free magnetic layers that interact through the Ruderman-Kittel-Kasuya-Yosida interaction. We determine analytically which collinear states along the easy axis that are stable, and establish numerically the phase diagram for when the system is in the STO mode and when collinear configurations are stable, respectively. It is found that the Rashba spin–orbit coupling can induce anti-damping in the vicinity of the collinear states, which assists the spin transfer torque in generating self-sustained oscillations, and that it can substantially increase the STO part of the phase diagram. Moreover, we find that the STO phase can extend deep into the antiferromagnetic regime in the presence of spin–orbit torques. Twenty years ago, it was theoretically proposed by Slonczewski1 and Berger2 that there could be exerted a torque on the magnetization in multilayer systems by passing a spin polarized current through the magnetic layers. This was coined the spin-transfer torque (STT) as the spin of the spin polarized current was transferred to the magnetic layer. This effect was observed experimentally3,4 a few years after the publications by Slonczewski and Berger, and spiked a lot of interest in the field as the magnetization could now be manipulated by electrical means, which is often advantageous practically compared to manipulation by magnetic fields. This torque could be used to switch the magnetization direction in one of the magnetic layers above some critical current4–6, which is of interest for writing techniques in memory technologies such as MRAM7,8 and racetrack memories9. The spin-transfer torque was also shown to induce a precession in the free magnetic layers10–12, which is now known as a spin torque oscillator (STO). The spin torque oscillator takes in a dc spin polarized current, and due to the precession in the free magnetic layers that causes an oscillation in the resistance through the giant magnetoresistance effect13,14, and the result is an ac current passing out of the multilayers. These alternating currents can have a wide range of frequencies, spanning the range of 100s of MHz to 100s of GHz8,15,16, and these frequencies are tunable by the magnitude of the applied current. Spin torque oscillators can exist in both antiferromagnetically17 and feromagnetically18 coupled magnetic layers, although it was noted in ref. 18 that they were unable to reproduce the antiferromagnetic STO phase predicted in ref. 17. Antiferromagnetic nano-oscillators were also recently considered in ref. 19. Another type of torque that has gained interest in magnetization dynamics in more recent years is the torque resulting from Rashba spin–orbit coupling (RSOC)20. This type of spin–orbit coupling occurs in materials with broken inversion symmetry, such as at the interface between two materials21. This inversion asymmetry causes an in-plane current flowing parallel to the interface to experience a magnetic field perpendicular to both the direction of the current and inversion asymmetry22. Rashba spin–orbit coupling has been shown to introduce interesting effects in many different areas of physics23, and is of particular interest due to the fact that the strength of the interaction can be tuned by gate voltages24,25. RSOC can, like STT, be utilized in magnetization switching, although RSOC is not the mechanism solely responsible for it26. Several works have considered the influence of spin–orbit torques on magnetic domain wall motion27–37. It has also been observed experimentally that the spin– orbit torque from RSOC can contribute to self-oscillations in STOs38. In this article, we show that RSOC can be used in metallic multilayer systems to substantially increase the size of the STO phase. Moreover, we discover an STO phase for two compensated antiferromagnetically coupled magnetic layers, which is a new result compared to e.g. Zhou et al. who could only find an STO phase Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Correspondence and requests for materials should be addressed to J.L. (email: [email protected])

Scientific Reports | 6:33845 | DOI: 10.1038/srep33845



Figure 1.  Illustrations of (a) the bulk geometry and (b) the thin film geometry. In both geometries, a fixed magnetic layer F0 is separated from two free magnetic layers F1 and F2 by a non-magnetic metallic material shown here in blue. A material illustrated in black, which is neighboring to F1 and F2 in (a) and the top film in (b), is present to get a strong Rashba spin–orbit coupling at the interface of the free ferromagnetic layers. A suitable choice for this material could be a heavy normal metal such as Au or Pt. In (b) the bottom film is a substrate which we assume does not induce any measurable interfacial spin–orbit coupling effects. To induce dynamics a current is applied in the x-direction which causes m1 and m2 to experience spin-transfer torques. In (a) a current is also applied in the y-direction to create significant RSOC effects on m1 and m2 due to the symmetry breaking in the x-direction. In the film geometry this is caused by the current in the x-direction as the symmetry breaking is in the z-direction. The free magnetic layers also interact through the RKKY interaction, while the distance to the fixed magnetic layer is chosen such that an RKKY interaction with this layer can be neglected. In (a) the material has an easy axis in the z-direction, while in (b) the material has an easy axis in the y-direction. for a ferromagnetic coupling in ref. 18, and Klein et al. could only find an STO phase for uncompensated antiferromagnetically coupled magnetic layers in ref. 17. We begin by setting up our model by utilizing the Landau-Liftshitz-Gilbert-Slonczewski (LLGS) equation, and then proceed by performing a Fourier transform of this equation to find when collinear states along the easy axis are stable. By comparing the new terms introduced by having RSOC present, we find that the spin–orbit torques effectively can be described by a modification of the Gilbert damping α​, to the extent where we can get an anti-damping term in the LLGS equation. To establish when we have an STO phase we solve the full LLGS equation numerically for different sets of experimentally relevant parameters, considering two possible geometries, and use the solutions to classify the phases. Lastly, we analyze the frequency spectrum of the STO phases by performing a Fourier transform of the solutions along different lines in the phase diagrams.


We will consider two different geometries where spin–orbit torques strongly influence the STO phase, and which display different behaviors. These geometries will henceforth be called the bulk and thin film geometries, and are illustrated in Fig. 1. Both geometries consist of a polarizing layer F0 and two free magnetic layers F1 and F2, all separated by a non-magnetic metal in order to reduce the exchange coupling and prevent magnetic locking. The main differences between the geometries is the direction of which the inversion symmetry is broken, and the current that is required to induce the spin–orbit coupling. In the bulk geometry the current in the y-direction induces spin–orbit torques (SOT) from RSOC, while the current in the x-direction induces the STT. Achieving current injection in two directions could potentially be challenging to achieve experimentally, but the possibility of separating the effects of STT and SOT by control of different current directions is an appealing concept that we here put forward in order to stimulate to experimental activity on such a setup. No such difficulty occurs in the thin film setup where there is only current injection in one direction. In the thin film geometry both STT and RSOC are caused by the same current in the x-direction, making it impossible so separate the effects from one another. In contrast, this is possible in the bulk geometry. The thin film geometry may also prove to be experimentally challenging, as it is fair to acknowledge the difficulties of electrically tuning the Rashba parameter in a well-defined manner in this geometry. The model for the dynamics of the magnetizations in the two free magnetic layers that we will study in this article is a variation of two coupled Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equations. This equation is given by R R STT ∂t mi = −γmi × (Heff i + H i − βmi × H i ) + αmi × ∂t mi + T i .

Scientific Reports | 6:33845 | DOI: 10.1038/srep33845



www.nature.com/scientificreports/ Here the Rashba field HRi is given by27 HiR =

(x / y ) j(x /y ) 1 α Rme Pi nˆ × ˆj , 2 eM s µ0 1+β


with α​R being the Rashba parameter, Pi(x /y ) the polarization of the current j(x/y) passing through Fi, and nˆ the direc-

tion of inversion asymmetry. The parameter β​is the non-adiabatic damping parameter of the itinerant electrons that describes the ratio of the inverse exchange interaction parameter 1/Jex and the spin-flip relaxation time. Lastly, the spin-transfer torques T STT are acting on mi are i T1STT = −

γ j x

2eM s µ0 d

T STT =− 2

m1 × (P0m0 − P1m2 ) × m1,

γ j x

2eM s µ0 d

P1m2 × m1 × m2 ,



with d being the thickness of the regions F1 and F2 in the x-direction, and P0 and P1 being the polarization of the current in the non-magnetic material in F0/N/F1 and F1/N/F2 respectively. Due to the symmetry of the geometry we can neglect the non-adiabatic spin transfer torque39. For the effective field Heff, we consider contributions from the RKKY interaction and magnetic anisotropy. The effective field then becomes Heff i =

2K J (mi ⋅ nˆ k ) nˆ k + mi . µ0 M s µ0 M s d


K is the anisotropy strength, nˆ k a direction along the easy axis (nˆ k = zˆ for the bulk geometry, and nˆ k = yˆ for the thin film geometry), J is the strength of the RKKY interaction, and the index i denotes the index of the free magnetization that is not mi (i = 3 − i). We now follow the procedure by Zhou et al.18 and consider when the collinear states of m1 and m2 along the easy axis are stable or not. We start off with the ansatz that there is a slight perturbation ui from the collinear state, such that mi = λi nˆ k + ui (λ​i =​  ±​1 ). Plugging this ansatz into (1) and performing a Fourier transform u i (t ) = ∫ u i (ω) exp( − iωt ) d ω /2π, we get a result that is on the form  1   = 0. ˆ ω + Vˆ )  u (A  u 2 


The matrices for the different geometries are given by 1 − iαλ1  0  , ˆ  A bulk =   0 1 − iαλ2 


1 + iαλ1  0  , ˆ  A film =   0 1 + iαλ 2 


λ 0   λ2 −λ1 (x )  −λ 0 Vˆ bulk = ω0  1  + ω J   + iω j P0  1   0 λ2  −λ2 λ1    0 0  y ( ) λ λ −1   0  ω(Ry )   P1  + P1 1 2  +  1 −λ1λ2   1 + β2   0 P (y )  2    P (y ) λ 0    − iβ  1 1  ,  0 P2(y ) λ2   

λ 0     − ω  λ 2 −λ1 + iω(x ) P −λ1 0 Vˆ film = − ω0  1 J j   0 λ 2   0  0 0 −λ λ  2 1   (x ) 0  λ λ −1   ω(Rx )   P1  + P1 1 2   −    2   1 −λ1λ 2   1 + β   0 P2(x )   P (x ) λ 0    + iβ  1 1  .  0 P2(x ) λ2   

Scientific Reports | 6:33845 | DOI: 10.1038/srep33845




www.nature.com/scientificreports/ Here, we have defined the frequencies ω​0  =​  2γ​K /μ 0 M s , ω​J  =​  γ​J /μ 0 M s d, ω(jx ) = γjx /2µ0 M s d and ˆ matrices, which can be attrib= γα Rme jx /y /µ0 eM s . In general there are also off-diagonal terms in the A uted to anisotropic damping terms such as spin pumping40. However, similarly to18 we neglect these terms to reduce the amount of parameters in our model. We now want to determine if any of the collinear states are stable when the frequencies and other constants are specified. This can be seen from the sign of the imaginary component of ω​; when the imaginary component is negative, exp(−​iω​t) is a decreasing function in time, while if the imaginary component is positive the function is exponentially increasing. Any small perturbation away from the collinear state is then unstable if Im(ω) > 0. The value of ω​can be determined from (6), as it can be written as an eigenvalue equation where ω​is an eigenvalue of ˆ = −A ˆ −1Vˆ . From this one can then find for what choice of parameters none of the collinear states the matrix W are stable. An STO phase will be localized within this region, but the entire region is not necessarily an STO phase. We here distinguish between the STO phase and a canted phase by considering the temporal evolution of the magnetoresistance, which is approximated by

ω(Rx /y )

R (t ) = R0 + ∆R1m0 ⋅ m1 + ∆R2 m1 ⋅ m2 .


If the magnetoresistance is oscillating in time, meaning at least m0 · m1 or m1 · m2 is oscillating, we have an STO phase. If m0 · m1 and m1 · m2 are both constant, we have a canted phase. Note that if m1 and m2 oscillate in-phase in a plane perpendicular to m0, such that m0 · m1 and m1 · m2 are constant, we still have a canted phase and not an STO phase even though there are oscillations in the individual magnetization components. To confirm our analytical results, we will also later establish fully numerically when the STO phase occurs. We now want to determine the physical effect that the Rashba spin–orbit coupling introduces by comparing it with other known terms. If we make the simplification of an equal polarization in the current causing the spin– orbit coupling in the free layers (P0 =​  P1 =​  P, P1(y ) = P2(y ) = P, P1(x ) = P2(x ) = P), this can be done without much effort. Note that for the bulk case it is not necessary to have the same polarization for both jx and jy, we could still achieve the same results with P1(y ) = P2(y ) = P′ by modifying α​R so that α​RP =​  α​′​RP′​. When performing this simˆ matrix. The influence plification in the polarization the matrix proportional to ω(Rx /y ) becomes very similar to the A of RSOC is then to renormalize α​ and ω​in the following manner: ω ⁎ = ω + ω(Rx /y )

α =

P , 1 + β2

αω + βω(Rx /y ) ω + ω(Rx /y )



1 + β2 P

. (13)

1 + β2

as ω(Rx /y )

We note that the imaginary components of ω​* and ω​are the same, is entirely real. The stability of a phase was determined by the imaginary component of its eigenvalue, therefore any effect that RSOC may have on shifting the borders between a stable and an unstable region must be seen from α​*. As ω​is in general complex, so is α​* (α​, however, is real). It is also noted that the product α⁎ω ⁎ = αω + ω(Rx /y ) P (1 + β2)−1 only has a shift in its real value. As ω​ and ω​* are complex and have the same imaginary component, the imaginary component of α​* must be non-zero if Re(α​*)  ≠​  α​. This imaginary component of α​* ensures that the imaginary component of α​ω​ is invariant under the transformation to α​*ω​*, but is otherwise of little interest as α​only appears in the product αω​. The effect of RSOC can then be found from the real part of α​*, which is found to be

(αRe(ω) + βω )(Re(ω) + ω ) + αIm(ω) . Re(α ) = (Re(ω) + ω ) + Im(ω) ⁎

(x / y ) P R 1 + β2

(x / y ) P R 1 + β2

(x / y ) P R 1 + β2





There is no restriction on the sign of ω​ or ω(Rx /y ) (ω(Ry ) can for example be negative by switching the direction of

the current jy), and Re(α​*) can as a consequence also take on negative values. The effect of RSOC can then be seen as a modification of the Gilbert damping parameter α​, even to the extent where it takes on negative values and becomes an anti-damping term in the LLGS equation. We note that the modification α​* depends on the eigenvalues ω​, which are complex functions of the system parameters. As the ansatz is a small perturbation from a collinear state, the modified value α​* is only valid in this state, and will change again in a manner that cannot be described by this framework if the perturbation is unstable and we move away from the collinear state. One special case that should be noted is when α​  =​  β​, for which Re(α​*)  =​  α​. This predicts that when α​  =​  β​, RSOC has no impact on whether a collinear state is stable or not. When α​  ≠​  β​, Re(α​*) has a maximum and a minimum when Im(ω) ≠ 0 and a singularity when Im(ω) = 0. These extremal points are found from ∂ Re(α⁎) = − (α − β) × ∂ω R


) + Re(ω)(Re(ω) + ω  + (Re(ω) + ω ) 

Im(ω)2 Re(ω) + 2ω R  Im(ω)2 


1 + β2


R 1 + β2

2 2

P R 1 + β2



=0 (15)

and are Scientific Reports | 6:33845 | DOI: 10.1038/srep33845



 1 + β2  Im(ω)2 ± Im(ω) Im(ω)2 + Re(ω)2 + Re(ω)2  ω R = −  .  P  Re(ω)


Which value is the maximum and minimum is determined by the sign of Re(ω​), and whether β​  >​  α​ or β​  ​  α​for the same set of parameters but with α​ 

Current driven spin-orbit torque oscillator: ferromagnetic and antiferromagnetic coupling.

We consider theoretically the impact of Rashba spin-orbit coupling on spin torque oscillators (STOs) in synthetic ferromagnets and antiferromagnets th...
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